Question 282618


Start with the given system of equations:

{{{system(3x-6y=-3,2x+4y=30)}}}



{{{2(3x-6y)=2(-3)}}} Multiply the both sides of the first equation by 2.



{{{6x-12y=-6}}} Distribute and multiply.



{{{3(2x+4y)=3(30)}}} Multiply the both sides of the second equation by 3.



{{{6x+12y=90}}} Distribute and multiply.



So we have the new system of equations:

{{{system(6x-12y=-6,6x+12y=90)}}}



Now add the equations together. You can do this by simply adding the two left sides and the two right sides separately like this:



{{{(6x-12y)+(6x+12y)=(-6)+(90)}}}



{{{(6x+6x)+(-12y+12y)=-6+90}}} Group like terms.



{{{12x+0y=84}}} Combine like terms.



{{{12x=84}}} Simplify.



{{{x=(84)/(12)}}} Divide both sides by {{{12}}} to isolate {{{x}}}.



{{{x=7}}} Reduce.



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{{{6x-12y=-6}}} Now go back to the first equation.



{{{6(7)-12y=-6}}} Plug in {{{x=7}}}.



{{{42-12y=-6}}} Multiply.



{{{-12y=-6-42}}} Subtract {{{42}}} from both sides.



{{{-12y=-48}}} Combine like terms on the right side.



{{{y=(-48)/(-12)}}} Divide both sides by {{{-12}}} to isolate {{{y}}}.



{{{y=4}}} Reduce.



So the solutions are {{{x=7}}} and {{{y=4}}}.



Which form the ordered pair *[Tex \LARGE \left(7,4\right)].



This means that the system is consistent and independent.



Notice when we graph the equations, we see that they intersect at *[Tex \LARGE \left(7,4\right)]. So this visually verifies our answer.



{{{drawing(500,500,-3,17,-6,14,
grid(1),
graph(500,500,-3,17,-6,14,(-3-3x)/(-6),(30-2x)/(4)),
circle(7,4,0.05),
circle(7,4,0.08),
circle(7,4,0.10)
)}}} Graph of {{{3x-6y=-3}}} (red) and {{{2x+4y=30}}} (green)